PROFIT MAXIMIZATION OF A KANBAN-BASED SUPPLY CHAIN. A Thesis. presented to. the Faculty of the Graduate School. at the University of Missouri-Columbia

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1 PROFIT MAXIMIZATION OF A KANBAN-BASED SUPPLY CHAIN A Thesis presented to the Faculty of the Graduate School at the University of Missouri-Columbia In Partial Fulfillment of the Requirements for the Degree Master of Science by PENG CHEN Dr. James S. Noble, Thesis Supervisor MAY 2015

2 The undersigned, appointed by the dean of the Graduate School, have examined the thesis entitled presented by Peng Chen, PROFIT MAXIMIZATION OF A KANBAN-BASED SUPPLY CHAIN a candidate for the degree of master of science, and hereby certify that, in their opinion, it is worthy of acceptance. Professor James S. Noble Professor Cerry Klein Professor Timothy C. Matisziw

3 ACKNOWLEDGEMENTS This thesis would never have been completed without the support from my faculty advisor, committee members, many other professors, and friends from the Industrial and Manufacturing Systems Engineering Department at University of Missouri-Columbia. I would like to express my deepest appreciation to my faculty advisor, Prof. James S. Noble. He taught me the way to conduct an academic research from scratch, by providing guidance for every step with great patience, from the literature review and topic selection, all the way to the model construction and optimization. More importantly, he inspired me to think independently thus gave me the confidence to better handle future research. I would like to thank Prof. Cerry Klein and Prof. Timothy C. Matisziw, my committee members, who provided me with many extremely valuable advice on model formulation, algorithm design, and thesis composition. Finally, I would also like to thank some department faculty who taught me the fundamental knowledge to conduct this research. Prof. Timothy Middelkoop provided me many practical programming tips, which made it more efficient to implement my study to large scale problems. Prof. Ronald McGarvey helped me to obtain deeper insights of the core stochastic components of the model. Prof. Cheng-Hsiung A. Chang taught me how to design factorial experiments, which I used to improve the performance of the search algorithm. ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS... ii LIST OF ABBREVIATIONS... v LIST OF FIGURES... vi LIST OF TABLES... viii ABSTRACT... xi 1 INTRODUCTION Kanban Puzzle Understand Kanban Kanban Model under JIT Philosophy Summary LITERATURE REVIEW Modelling Approach Solution Methodology System Types Summary MODEL DEVELOPMENT Assumption Parameter Decision Variable Other Notation Objective Function Explanation Iteration Constraint SOLUTION METHODOLOGY Solution Vector Objective Function Search Space Perturbation Cooling Schedule iii

5 4.6 Stopping Rules SA Pseudocode RESULT Kanban Optimization Model Analysis CONCLUSION AND FUTURE RESEARCH Conclusion Future Research REFERENCE iv

6 LIST OF ABBREVIATIONS BOM stands for Bill of Material, 21 CONWIP stands for Constant Work-in-Process, 5 GA stands for Genetic Algorithm, 15 JIT stands for Just-in-Time, xiii MRP stands for Material Requirements Planning, 10 PK stands for Production Kanban, 19 SA stands for Simulated Annealing, xiv, 7 TPS stands for Toyota Production System, xiii WIP stands for Work-in-Process, 1 stands for Withdrawal Kanban, 18 v

7 LIST OF FIGURES Figure 3-1 Workflow of a double-kanban-based supply chain Figure 3-2 Demand flow Figure 3-3 Demand flows through customer shop and 1st supplier Figure 3-4 Inventory plot of the customer shop Figure 3-5 Inventory plot of the 1st supplier s warehouse Figure 5-1 One-customer-shop and one-supplier (C-S) supply chain Figure 5-2 Main effects plot (left) and interaction plot of tf and β of perturbation one SA on the C-S supply chain Figure 5-3 Interaction plot of ti, tf, β of perturbation one SA on the C-S supply chain 58 Figure 5-4 Interaction plot of tf, α and β of perturbation one SA on the C-S supply chain Figure 5-5 Interaction plot of tf, β and π of perturbation one SA on the C-S supply chain Figure 5-6 Search path of perturbation one SA on the C-S supply chain Figure 5-7 Main effects plot (left) and interaction plot of tf, β and π of perturbation two SA on the C-S supply chain Figure 5-8 Interaction plot of ti, tf, β and π of perturbation two SA on the C-S supply chain Figure 5-9 Interaction plot of tf, α, β and π of perturbation two SA on the C-S supply chain Figure 5-10 Search path of perturbation two SA on the C-S supply chain Figure 5-11 One-customer-shop & two-supplier (C-S-S) supply chain Figure 5-12 Main effects plot (left) and interaction plot of tf, β and π of perturbation one SA on the C-S-S supply chain vi

8 Figure 5-13 Interaction plot of ti, tf, β and π of perturbation one SA on the C-S-S supply chain Figure 5-14 Interaction plot of tf, α, β, π of perturbation one SA on the C-S-S supply chain Figure 5-15 Searching path of perturbation one SA on the C-S-S supply chain Figure 5-16 Main effects plot (left) and interaction plot of β and π of perturbation two SA on the C-S-S supply chain Figure 5-17 Interaction plot of ti, β and π of perturbation two SA on the C-S-S supply chain Figure 5-18 Interaction plot of tf, β and π of perturbation two SA on the C-S-S supply chain Figure 5-19 Interaction plot of α, β and π of perturbation two SA on the C-S-S supply chain Figure 5-20 Searching path of perturbation two SA on the C-S-S supply chain Figure 5-21 Optimum t increases with G Figure 5-22 f(t) under different G Figure 5-23 Various demand rate Figure 5-24 Various time limit Figure 5-25 Various reliability Figure 5-26 Various inventory Cost Figure 5-27 Various customer rejection cost Figure 5-28 Various price Figure 5-29 Make a more scientific decision vii

9 LIST OF TABLES Table 5-1 Parameters of the C-S supply chain Table 5-2 K1 experiment on the C-S supply chain Table 5-3 SLc experiment on the C-S supply chain Table 5-4 t experiment on the C-S supply chain Table 5-5 Search space of the C-S supply chain Table 5-6 Factor experimental levels of SA parameter Table 5-7 Perturbation one SA parameter experiment on the C-S supply chain Table 5-8 ANOVA table of perturbation one SA parameter experiment on the C-S supply chain Table 5-9 Parameter of perturbation one SA on the C-S supply chain Table 5-10 Result of perturbation one SA on the C-S supply chain Table 5-11 Perturbation two SA parameter experiment on the C-S supply chain Table 5-12 ANOVA table of perturbation two SA parameter experiment on the C-S supply chain Table 5-13 Parameter of perturbation two SA on the C-S supply chain Table 5-14 Result of perturbation two SA on the C-S supply chain Table 5-15 Compare the SA results using perturbation one and perturbation two on the C-S supply chain Table 5-16 T-test between the SA results of perturbation one and perturbation two on the C-S supply chain Table 5-17 Optimum allocation of the C-S supply chain Table 5-18 Parameters of the C-S-S supply chain Table 5-19 K1 experiment on the C-S-S supply chain viii

10 Table 5-20 K2 experiment on the C-S-S supply chain Table 5-21 SLc experiment on the C-S-S supply chain Table 5-22 SL1 experiment on the C-S-S supply chain Table 5-23 t experiment on the C-S-S supply chain Table 5-24 Search space of the C-S-S supply chain Table 5-25 Perturbation one SA parameter experiment on the C-S-S supply chain Table 5-26 ANOVA table of perturbation one SA parameter experiment on the C-S-S supply chain Table 5-27 Parameter of perturbation one SA on the C-S-S supply chain Table 5-28 Result of perturbation one SA on the C-S-S supply chain Table 5-29 Perturbation two SA parameter experiment on the C-S-S supply chain Table 5-30 ANOVA table of perturbation two SA parameter experiment on the C-S-S supply chain Table 5-31 Parameter of perturbation two SA on the C-S-S supply chain Table 5-32 Result of perturbation two SA on the C-S-S supply chain Table 5-33 Compare the SA results using perturbation one and perturbation two on the C-S-S supply chain Table 5-34 T-test between the SA results of perturbation one and perturbation two on the C-S-S supply chain Table 5-35 Optimum allocation of the C-S-S supply chain Table 5-36 Optimum t increases with G Table 5-37 f(t) under different G Table 5-38 Various demand rate Table 5-39 Various time limit Table 5-40 Various reliability Table 5-41 Various inventory Cost ix

11 Table 5-42 Various customer rejection cost Table 5-43 Various price Table 5-44 Positive Changes Table 5-45 Negative Changes x

12 ABSTRACT Kanban production planning and control has been intensively studied worldwide since early 1980s. However, most modern research treats Kanban as an isolated concept. It is too often just modeled as containers holding production parts. The problem is that Kanban is not an isolated concept, but it is a key component integrated in the Toyota Production System (TPS). The purpose of implementing Kanban is to achieve Just-in-Time (JIT) production. There is no way Kanban alone could accomplish JIT without level production. Level production is based on a flexible operation interval. Therefore, we need to incorporate the level production aspect of JIT into Kanban modelling. A good way to do this is to include operation time interval as a decision variable, which, however, is a missing criterion in most of the Kanban research. There is a misunderstanding that JIT means zero inventory. The inventory level of a JIT system is directly associated with the number of Kanban, but so is the customer service level. In order to find the optimum number of Kanban, we need to make a tradeoff between the inventory cost and the customer service level. Profit is a good performance measurement to guide this trade-off because it can clearly compare these two issues. However, most of the existing research uses cost as the objective function, based on the assumption that inventory always turn into sales. The above reasons motivate us to build a new Kanban model. In this paper, we propose a stochastic method to describe a multi-stage Kanban-based supply chain. As indicated before, we include the operation time interval as a decision variable, and use the overall supply chain profit as the objective function. Beyond that, we also introduce some xi

13 practical considerations to make our model more robust, such as the opportunity cost associated with unsatisfied customer demands, the transportation risk, and the production risk. After developing the model, we implement a Simulated Annealing (SA) technique to efficiently find the optimum allocation of the Kanban-based supply chain. Finally, we conduct sensitivity analysis, which not only proves the robustness of our model, but also provides us with a tool to make fact-based decisions in response to unexpected external changes. This enables us to partially recover the profit when the changes are negative or take a better advantage of changes when they are positive. xii

14 1 INTRODUCTION 1.1 Kanban Puzzle Kanban is a well-known concept in the field of production planning and control. In fact, it has been applied not only to manufacturing, but also many other industries, such as hospitals, restaurants and amusement parks, wherever workflow control is a key factor to success. However, not all were able to implement Kanban in the right way (Hopp et al., 2004). Those who understood and implemented Kanban correctly achieved lower Workin-Process (WIP) level, shorter lead time, more stable throughput, better product quality, and better customer service. As a result, they improved their profits. On the other hand, those who only applied Kanban on a superficial level suffered from inventory shortage, longer lead time, and degraded customer service. They invested huge amounts of money and effort in adapting their business to Kanban, but Kanban paid them nothing. Therefore, it is significant to understand the real working logic behind Kanban, then use the correct logic to develop a tool to facilitate Kanban implementation. 1.2 Understand Kanban The Big Picture Originally, Kanban is just a component of TPS. It is not even a direct component, but a second-layer component. TPS is directly supported by two pillars: autonomation and JIT production. Autonomation is about making full use of people s capacity. JIT is about setting up a unique production environment and logic so that people are working towards the right direction. JIT has two components: level production and Kanban. Level production is a prerequisite to implementing Kanban, and level production itself is based 1

15 on reduced setup time and a flexible transportation period. After accomplishing level production, Kanban can then be applied as a media to transfer flexible demand information from downstream to upstream while accompanying requested material from upstream to downstream (Sugimori et al., 1977). Since autonomation is beyond the scope of this research, we only focus on JIT aspect of TPS. We also believe that the JIT scope is big enough for us to fully explore the functionality of Kanban mechanism Just-in-Time Production The ultimate goal of JIT is pattern matching. In a JIT system, every upstream production activity should be closely matched to (triggered by) its downstream consumption activity. Ideally, when one unit of final product in the most upstream facility is consumed, all the downstream facilities should work together only to produce one new unit of that just consumed final product, and nothing more. Therefore, no waste would ever happen. In order to achieve this goal, two things need to happen: 1) the upstream knows what is happening at the downstream. 2) the upstream has the ability to respond to what is happening at the downstream. Kanban helps the upstream to know what is happening at the downstream. In a JIT system, a product in a downstream facility is always attached with a piece of Kanban. Ideally, at the moment just before the product is consumed, its Kanban will be detached and sent to the upstream facility. The Kanban tells the upstream facility that a product has just been consumed and it needs to supply a new product to the downstream. If we imagine the activity information at the downstream is a continuous signal, then the actual information that Kanban carries to the upstream is a discretized signal. Therefore, the more frequently the Kanban runs, the more timely information it conveys. Also, Kanban 2

16 is part-specific, which means that in a multi-product manufacturing environment the upstream facility can use the Kanban to identify which product the downstream facility is asking for. Level production enables the upstream to respond to what is happening at the downstream. Let s say there are two kinds of product in a brand new JIT-based business, product A and product B. Its first three customers just did some shopping at the downstream facility. Customer 1 bought a product A, Customer 2 bought a product B, and Customer 3 bought a product A. According to the JIT protocol, the upstream facility should have received two Kanban A and one Kanban B. Let s say there are two options for the upstream facility to arrange the production. Option one is to first produce two units of product A then produce one unit of product B. Option two is to first produce one unit of product A then produce one unit of product B then produce another one unit of product A. Which one is better? Ideally, in a JIT system, the upstream facility should choose option two. Here is the reason. The shopping record of the first three customers is the only data you can use to predict the future customer behaviors. Therefore, chances are, at the downstream facility, Customer 4 will want to buy a product A, Customer 5 will want to buy a product B and Customer 6 will want to buy a product A. If this is the case, then production option two can use less time to supply one unit of product B to the downstream facility thus reduce the odds to lose Customer 5. Option two matches the activity patterns of the two facilities closer! The only problem is that production option two seems a little inconvenient because the upstream facility has to switch back and forth to conduct different productions. However, the JIT practitioners don t really care about convenience. They care about financial performance. As long as the frequent switches won t increase cost nor decrease 3

17 income, inconvenience is totally acceptable. This sounds impossible, but it is achievable. Toyota Company, for example, has the record to allow the switches to occur in every ten minutes (sometimes even as short as three minutes) without increasing its manufacturing cost. When a system can achieve such kind of highly frequent and finance friendly production switches, we say it is able to conduct level production. It is meaningless for the upstream to know what is happening at the downstream if it is unable to respond efficiently. As state before, the mission of Kanban is to convey downstream information to upstream as accurately as possible. Then, if the upstream can conduct level production, it can match its activity to the downstream according to the Kanban information. However, if the upstream cannot conduct level production, then even it has the target to aim, it does not have the ability to match. Therefore, we say that being able to conduct level production is a prerequisite of implementing Kanban mechanism. Idealism is expensive, and often impossible to achieve. We want to run Kanban as frequently as possible. But there is a transportation cost, such as gas cost and depreciation cost. Transportation cost forces us to set larger buffers, which, in the case of JIT, means more Kanban. But Kanban needs to be attached on products. Therefore, more Kanban means more products in the buffer. More products in the buffer means higher inventory cost. Then how about cutting the buffer size? Well, that might compromise the customer service level, which hurts the income, thus the profits. JIT is an ideal concept. In reality, however, we need to make trade-off. Trade-off requires a unified performance measure, around which we can build a JIT/Kanban model. A robust model helps us make reasonable trade-off and find the optimum Kanban allocation. OK, Kanban is not just an information carrier. It also provides another benefit. However, this benefit has nothing to do with Kanban modelling, because it is hard to 4

18 measure and quantify. More all, this benefit will naturally appear as long as you deploy the Kanban system reasonably - Kanban provides an explicit way to constraint WIP level. According to Kanban mechanism, the WIP level will never excess the pre-determined Kanban quantity. Therefore, the WIP level in a Kanban system will always be relatively low and stable, which extremely improves the working efficiency of the shop floor operators. The reasons behind that can be both physical and mental, pretty much a human-factor issue, which is beyond the scope of this research, but still worth mentioning. The most famous application of this Kanban benefit is the invention of CONWIP Constant Work-in-Process method. 1.3 Kanban Model under JIT Philosophy Most of the modern research treat Kanban as an insulated concept. According to Japanese translation, Kanban means card or container. Therefore, Kanban is too often just modeled as the containers holding production parts. And that is the problem. Kanban is not an insulated concept, but it is an organic component integrated in JIT. First of all, the purpose of implementing Kanban is to achieve Just-in-Time production. But there is no way Kanban alone could accomplish JIT without the support of level production. Level production is based on flexible operation interval, which can be easily changed according to market demands. Therefore, we need to incorporate the level production aspect of JIT into Kanban modelling. And a good way to do this is to include operation interval as a decision variable, which, however, is a missing criterion in most of the Kanban models. Secondly, there is a misunderstanding that JIT systems have zero inventory. If that were true, when customers arrived, JIT practitioners might just conjure their products instantly from nowhere. Apparently, it is not possible. JIT systems do have inventory, 5

19 except that all the products in their inventory are attached with and controlled by some kind of Kanban, ready to be pulled. The inventory cost of a JIT system is directly associated with the number of Kanban. The mission of Kanban is to carry accurate periodical demand information to support the level production. The more Kanban you have, the more accurate information you can provide to the production shop floor, and the higher customer service level you can obtain. However, more Kanban means higher inventory cost. Will this customer service level margin outweigh the inventory cost margin, both of which caused by additional Kanban? In order to answer this question, we need to put the customer service level and the inventory cost under a unified performance measure. Although many Kanban models use system cost as their objective functions, we don t think cost alone could reflect the customer service level. Therefore, we prefer to use profit as the performance measure, which is significantly influenced by both the customer service level and the inventory cost. The reasons stated above is exactly what motivates us to conduct this research. 1.4 Summary In this chapter, we have introduced how a Kanban system works under JIT philosophy, and also addressed the problems that motivate us to conduct this research. As stated, the purpose of this work is to build a more reasonable model for the Kanban system and find its optimal allocation. Therefore, we also need a literature review of state-of-art studies on the Kanban topic, and it is the content of chapter 2. In chapter 3, we present the objective function of our model. Then we elaborate the model by further explaining the objective function. Readers should get the whole picture of our model after 6

20 reading through the entire chapter. In chapter 4, we introduce Simulated Annealing (SA) as the search algorithm for our model. The results of our study are demonstrated in chapter 5. In chapter 6, we conclude our study and propose several future research opportunities that could stem from our work. 7

21 2 LITERATURE REVIEW As stated in the last chapter, the goal of our research is to build a Kanban model that meet two major criteria. First, it should provide the time flexibility required by level production. Second, its objective function should use such a performance measurement so that the trade-off between inventory cost and customer service level in a Kanban system can be made. Throughout this literature review, we pay close attention to the existing research that either include these criteria or provide the possibility to be extended with these ideas. With this goal in mind, we review the research published in the past 10 years from three different perspective: modelling approach, solution methodology, and types of modelled Kanban system. 2.1 Modelling Approach Analytical Model In Ramanan et al. (2003) modelled a Kanban system manufacturing different parts. Different part-types are put into different kinds of Kanban containers. And there are several containers for each part-type. Their decision variable is the processing sequence of the containers. The objective is to minimize the makespan of the containers. However, in our opinion, the process sequence of different parts in a Kanban system should primarily be determined by the downstream ordering sequence (or pull sequence), not the length of any makespan. Moreover, for a system which should be able to conduct level production, the production sequence won t make a big difference on the overall makespan anyway. 8

22 Prasad et al. (2006) did a similar research. Their decision variable is also the sequence of containers. In addition to minimize the average makespan of both containers and part types, they also aimed to minimize the standard deviation of between different parts. Therefore, their research is a multi-objective optimization problem. Similarly, we still don t think the makespan is an appropriate performance measurement for a Kanban/JIT system. Also, we don t see a big benefit for a Kanban/JIT system to have a minimized fluctuation of the makespan between different parts. Wang et al. (2006) developed a model to describe a multi-stage supply chain. In their research, the parts in the most upstream plant are related to raw material inventory, the parts in all the intermediate plants are related to WIP inventory, and the parts in the most downstream plant are related to finished good inventory. Their parameters include the total demand for the supply chain, the production rate of each plant, the cost of each item, the time between successive shipments for each stage, the inventory holding cost for each plant, the setup cost for each plant and each, the cycle time, the uptime and down time of each plant. Their decision variables contains the total production during each cycle time, the Kanban size at each plant, the number of Kanban at each plant, the number of orders or shipments placed at each plant, and the number of shipments placed at each plant s uptime. They use these parameters and variables to derive the average inventory level at each plant and the corresponding inventory and manufacturing cost. Their objective is to minimise the total cost. Our major concern is their assumption. They assumed they that the supply chain has a fixed total demand. So they divided this total demand into different periods, then split the periodical demands onto each Kanban and shipment. As a consequence, they were forced to assume that all the products manufactured by this Kanban system can be completely consumed by the total demand rate and all turn into actual income. Therefore, 9

23 they had their reason to only focus on the cost. However, this assumption sounds more like a MRP, not JIT. It is well know that the major problem of MRP is excessive WIP, which is caused by inaccurate demand estimation. The users of MRP had already realized that they were left behind by Japanese manufacturers in early 1980s, who used JIT as their tools. The reason is that MRP tries to predict demand, which is highly unpredictable, while JIT tries to match ready happened demand as swiftly as possible, which is much more realistic. Therefore, when we are using the assumption of a fixed demand to model a production system, we are actually modelling a MRP, not a JIT, no matter whether your model involves Kanban or not. What matters is the philosophy behind the model, not what you name your parameters and variables. Now that the assumption of a fixed demand leads to a fake Kanban/JIT model, we had better not use it, which leads to another issue if the demand is dynamic, there is no way we can ensure that the finished products can all turn into profits. If not all the finished products can become profits, only focusing on the cost can be shortsighted. As stated in chapter one, we need to compare the inventory cost margin and the customer service level related income margin, both of which attribute to additional Kanban. In a word, when there is no guarantee that all inventory can become sales, we should use profit instead of cost as the performance measure for our production system. But Wang et al. (2006) do provides many valuable considerations of a production system. For example, their parameters contain the time between successive shipments and the cycle time, which reflect the operation frequency issue, which can be used to model the level production aspect of a JIT system after modification. In addition, they also considered uptime and downtime of the plants, which reminds us of the potential risks in a production system that we should include in our own model. 10

24 2.1.2 Markov Chain & Queuing Network Tardif et al. (2001) and Shahabudeen et al. (2008) used Markov process to model an adaptive Kanban system. There their four decision variables in their model, the number of Kanban, the number of backup Kanban, the backup Kanban release limit and the back Kanban capture limit. Every time customer demands arrive, the system s inventory level and backorder level be compared to the two limits in order to decide whether the system should add more Kanban into the circulation or reduce the Kanban number or leave it alone. In this way, the number of Kanban in their model is dynamic. Their objective is to find the optimum values for the decision variables so that the sum of inventory cost and backorder cost can be minimized. The idea of making the number of Kanban dynamic is very creative. However, we also have some concerns about this research. First of all, the dynamic number of Kanban will compromise the inherent feature of a stable WIP level in a Kanban system. The success of CONWIP has ready proven the significance of a stable WIP level in a production system. Therefore, if the backup Kanban is released or retrieved too often, the upper limit of the WIP level in the adaptive Kanban will become unstable, which may cause chaos in the shop floor. Secondly, we still believe that the flexibility of a real Kanban/JIT system should be provided by high-quality level production, not a dynamic Kanban number. A changeable number of Kanban could a superficial rescue method to avoid inventory shortage or excess at the time, but it won t solve the problem from the root in the long run. Again, in a Kanban system. The shortage or excess of Kanban is merely a symptom of poor level production. In order to be relieved from the symptom, we have to cure the core disease first. If level production has been achieved, an optimum fixed number of 11

25 Kanban should work just fine. Let alone a fixed number of Kanban offers a huge bonus by providing a stable WIP, so we d better not touch it. Tardif et al. (2001) and Shahabudeen et al. (2008) do give us an idea of how to capture the performance of customer service level. In their study, they used backorder cost. The higher the customer service level is, the lower the backorder cost. However, the backorder cost is not easy to decide. How much do you want to punish yourself if you cannot fulfil a customer s order? Well, this is actually an issue of opportunity cost, which has never been easy to decide. Especially in our case, when disappointed customers left because you run out of their desired products, you never know they will turn their backs to you forever or just come back and try their luck again tomorrow. If the former happens, you should punish yourself hard. If the latter happens, the backorder shouldn t cost you at all. The problem is that you never know. In a word, backorder cost is a vague measurement of customer service level. It may serve as an assistant measurement, but we shouldn t depend on it. We want something more direct, something happens right at the moment when a customer leaves because of your inventory shortage. Yes, you are losing the money that customer already decided to pay. That s your direct measurement. Matzka et al. (2012) used a queuing network to model the Kanban system. Their decision variable is the number of Kanban. Through the queuing network, the number of Kanban can be mapped to a final customer service level. A big advantage of this research is that it is very generic. They didn t provide an explicit objective function. But they did clearly state the close relationship between the number of Kanban and the customer service level. Moreover, in the process of deriving the customer service level, the inventory information can also be extracted from the queuing network. Last but not least, they included operation interval as a very important parameter, and explicitly conducted a solo sensitivity analysis on this time parameter to prove its significant role in deciding the 12

26 relationship between Kanban and service level. We believe that the motivation behind their research is very close to ours, and there is a big possibility that their model can be extended to meet the two criteria we listed at the beginning of this chapter. In Al-Tahat et al. (2005) s and Al-Tahat et al. (2012) s queuing network Kanban models, the decision variables are number of Kanban, utilization rate of production process, and the number of servers in each work station. Their objective is to minimize the total cost, which consists of raw material cost, manufacturing cost, transportation cost, and inventory holding cost. Our concern about this research is similar as the one we have with Wang et al. (2006) s research, it does not include the service level aspect of a Kanban/JIT system, which may lead to biased solution. However, treating the number of servers as a variable is a good idea, which could be used in our own model Simulation Shahabudeen et al. (2002) and Shahabudeen et al. (2003) ran simulations to mimic the working process of a single-kanban system and a double-kanban system respectively. The primary settings of their simulations are: multiple products, multiple workstations, stochastic customer demand, and stochastic processing time. By changing the settings of Kanban number and Kanban size, they tried to find the optimum combination based on which the simulation results contributed to a bigger aggregated demand fulfil rate and a smaller aggregated WIP level. We strongly agree with their objective, because their demand fulfil rate matches our customer service level and their WIP level can be modified to our inventory cost. However, one of their assumption is too strong for us. They assumed there was no substantial distance between the workstations and the Kanban detached from the downstream can be carried to the upstream immediately. 13

27 Nomura et al. (2004) simulated a double-kanban system with a modular approach. In their simulation, all the time related parameters can be deterministic or stochastic, and can follow any kind of distribution. From the simulation results, they derived demand fulfil rate, utilization rate of the resources, and average flow time. They also analyzed the structure of the flow time how much was spent on processing and how much was spent on inventory or WIP. Their research provided a generic Kanban system simulation methodology. Murino et al. (2010) simulated the work flow of a supermarket which implemented Kanban method. In their research, there are three kinds of Kanban, which had different urgent levels. The Kanban with the highest urgent level should always be dealt first. Their decision variables are the numbers of these three kinds of Kanban. Their objective function is to minimize the sum of the inventory cost and backlog cost. Similar as the one to Tardif et al. (2001) s research, our concern on this work is also about the backlog cost they used in their objective function, which could be a vague measurement of customer service level Meta Model Savsar et al. (2000) combined factorial design, simulation and neural network to develop a Kanban system modelling method, which provided three key features: 1) highquality search space, 2) accurate modelling result, and 3) significantly reduced computation time. Their performance measures included both WIP level and demand delay time. They also suggested a way to use a weighed function to put these two measures under the same unit. Our question about this research is how to decide the weights. Hou et al. (2011) did a similar work, but they used a regression model instead of the neural network. 14

28 2.2 Solution Methodology Among the literature which explicitly stated their solution methodologies, nearly half of them used traditional optimization algorithms, while the other half implemented meta-heuristics Traditional Optimization Savsar et al. (2000) used exhaustive enumeration. Greedy heuristic was chosen by Tardif et al. (2001). In Wang et al. (2004) and Wang et al. (2006) s studies, branch and bound algorithm was implemented. Al-Tahat et al. (2012) used dynamic programming algorithm to find their best solution Meta-heuristics Shahabudeen et al. (2002), Ramanan et al. (2003) and Shahabudeen et al. (2008) embedded their models into SA algorithms to search for the optimum solution. While in Gaury et al. (2000), Köchel et al. (2002), Prasad et al. (2006), Shahabudeen et al. (2008) and Hou et al. (2011) s studies, their models were incorporated into Genetic Algorithms (GA) to research the objectives. 2.3 System Types Various production systems have been studied in the literatures, from the simplest single-product and single-stage systems to the most complicated multi-product and multistage systems Single-product and Single-stage Tardif et al. (2001), Wang et al. (2004), Shahabudeen et al. (2008) and Matzka et al. (2012) studied single-product and single-stage systems. 15

29 2.3.2 Single-product and Multi-stage Gaury et al. (2000), Savsar et al. (2000), Chan et al. (2001), Al-Tahat et al. (2005), Wang et al. (2006), Prasad et al. (2006), Takahashi et al. (2007), Pettersen et al. (2009), Hou et al. (2011) and Al-Tahat et al. (2012) studied single-product and multi-stage systems Multi-product and Single-stage Murino et al. (2010) studied multi-product and single-stage system Multi-product and Multi-stage Chan et al. (2001), Shahabudeen et al. (2002), Ramanan et al. (2003), and Nomura et al. (2004) studied multi-product and multi-stage systems. 2.4 Summary The two most popular performance measures of a Kanban system are operation cost related measure and customer service level related measure. Researchers who assume that all the inventory would turn into sales tend to only focus on the operation cost. But researchers who really used JIT philology to guide their studies often took the customer service level related measure into consideration as well. However, the latter researchers struggled finding a reasonable way to put these two performance measures under the same unit. Some of them tried to use backorder cost to match the customer service level related measure to the operation cost related measure, but the backorder cost was too vague to be decided. Others tried to put these two measures in a unified weighted function, but the weights were also too difficult to be found. Therefore, we conclude this presents a research gap. In our research, we use the periodical profit as the objective 16

30 function, which includes both of the two measures in a very reasonable manner. This literature review also confirms our statement in chapter 1 that few researchers use the operation frequency (or operation interval) as a decision variable, so this is second research gap to fill. We also collect some very valuable Kanban modelling considerations from this review, such as the uptime and downtime of the plant, the transportation issue, the number of servers, and opportunity cost. These factors are all included in our research. We also find a trend that researchers have been building more and more complicated models since meta-heuristics came about. The reason might be that the simplified models of Kanban systems had already been exploited in the 1980s and 1990s JIT movement. Therefore, modern researchers are required to build more sophisticated models which describe the Kanban system more accurately. However, these complicated models always end up as non-linear or NP hard, which are impossible to solve using traditional analytical methods. Therefore, they resorted to meta-heuristics. As stated in chapter 1, we want to construct a comprehensive Kanban model constructed strictly under a JIT philosophy. The proven effectiveness of meta-heuristics give us the confidence that our model can be optimized. Hence, the real challenge is to build a good model. 17

31 3 MODEL DEVELOPMENT Matzka et al. (2012) modeled a single-stage Kanban system as a queueing network with synchronization stations. Their aim is to determine the optimal number of production Kanbans, and thus the buffer size that guarantee a given service level. Also, they included time interval as one of the parameters. The central part of their model is a Markov transition matrix, with its states denoting the number of finished products and waiting demands in each supplier. We extend their model in the following ways. First of all, we embed more stochastic factors into this model, such as transportation and production risks. In this way, our model is able to describe the Kanban system more closely and become more robust. Secondly, we extend this model from two-stage to multi-stage by clarifying its interfaces, so that sequential stations can be connected together with these interfaces. Last but not least, we direct all the variables and interactions of the model to a single financial objective function: the overall profit generated from the supply chain. This objective serves as the ultimate goal for guiding the tradeoff of all the sub-objectives. The flow chart of the multi-stage Kanban system that we want to model is shown in Figure 3-1, in which the red dashed lines represent information (Kanban) flow and the green weighted lines represent material (products/sub-products) flow. And the numbers in the figure correspond to the following comments: 1. At the customer shop, each unit of final product is attached with one piece of Withdrawal Kanban (). 2. When one unit of final product is purchased by a customer, its be will be detached and collected in the withdrawal Kanban post. 18

32 3. Periodically, there will be trucks, which travel between the customer shop and the 1st supplier, arriving, unloading final products, loading s and taking them to the 1st supplier. 4. At the 1st supplier, the final products are stored in the output inventory, where each unit of final product is attached with one piece of production Kanban (PK). 5. When the trucks stated in 3 arrive at the 1st supplier, they unload s and load final products, and there are two kinds of possibilities. If the number of finished products in the supplier s output buffer is equal or greater than the number of s, then each piece of will be attached to one unit of product and together loaded back in the trucks, and taken to the customer shop. Otherwise, the excessive s will have to wait in the waiting demand queue unit another unit of product is available. 6. When the product is attached with a, its PK has to be detached and taken to the supplier s plant. 7. The plant processes its products according to the number of PKs collected from the output inventory. One piece of PK corresponds to one unit of production demand. No production Kanban, no production. 8. In order to manufacture products, the plant need to take raw materials from its warehouse. The warehouse functions similarly as the customer shop does, except that its customer is the plant, and it does not charge money from the plant. Note that there s no Kanban circulating between the plant and the warehouse. 9. When one unit of product is manufactured, it will be attached with one piece of PK and carried to replenish the output inventory. 10. All the other intermediate suppliers are omitted here, all of which work in the similar way as the 1st supplier does. The only difference is that they process products at various levels along the supply chain. 11. At the most upstream supplier, the plant takes raw material directly from natural resource site. Hence, there is no warehouse in this last supplier. 19

33 Customer Station The 1 st Supplier 5 4 PK PK PK 9 7 PK PK PK PK PK PK PK PK PK 6 PK The Most Upstream Supplier PK PK PK PK PK PK 11 PK PK PK PK PK PK PK Natural Resource Figure 3-1 Workflow of a double-kanban-based supply chain 20

34 3.1 Assumption Our Kanban model is built under the following assumptions: 1. The supply chain only manufactures one single final product type. 2. The bill of material (BOM) is simplified. The manufacturing route is a single flow line. Every material ratio between a part and its sub-parts is one. 3. There are parallel machines in each plant. Each machine s uptime and downtime follow exponential distributions. 4. Every part is only processed by one machine in a plant, and sent to the downstream for further process. Every single process time is shorter than the synchronized operation time interval of the entire system. 5. There are trucks in each stage of the supply chain for transportation. Each truck s uptime and downtime follow exponential distributions. The time needed for a round-trip in each stage is shorter than the synchronized operation time interval of the entire system. 6. The most downstream customer demand follows a Poisson distribution. 3.2 Parameter System Parameters D= Demand rate comes into customer shop per unit time N= Number of suppliers in the Kanban system c Tc = Capacity of each truck transporting products between the customer shop and the 1 st supplier λ Tc = Failure rate of the each truck transporting products between the customer shop and the 1 st supplier per unit time β Tc = Repairing time of each unserviceable truck transporting products between the customer shop and the 1 st supplier c Ti = Capacity of each truck transporting products between the i th supplier and the (i + 1) th supplier 21

35 λ Ti = Failure rate of the each truck transporting products between the i th supplier and the (i + 1) th supplier per unit time β Ti = Repairing time of each unserviceable truck transporting products between the i th supplier and the (i + 1) th supplier c Mi = Capacity of each machine at the i th supplier per unit time λ Mi = Failure rate of each machine at the i th supplier per unit time β Mi = Repairing time of each unserviceable machine at the i th supplier Financial Parameters P C = Unit price of the product at customer shop C c = Unit cost of product at customer shop R c = Unit demand rejection cost at the customer shop H c = Unit inventory holding cost at customer shop per unit time T c = Unit depreciation and repairing cost of the trucks transporting products between the customer shop and the 1 st supplier per unit time G c = Gas cost per round trip per truck that transports products between the customer shop and the 1 st supplier P i = Unit price of the product at the i th supplier C i = Unit raw material cost for manufacturing the product at the i th supplier R i = Unit demand rejection cost at the i th supplier HO i = Unit output inventory holding cost of the i th supplier per unit time HI i = Unit input inventory holding cost of the i th supplier per unit time T i = Unit depreciation and repairing cost of the trucks transporting products between the i th supplier and the (i + 1) th supplier per unit time G i = Gas cost per round trip per truck that transports products between the i th supplier and the (i + 1) th supplier M i = Unit operation cost of the machines at the i th supplier per unit time 22

36 Note: C c = P 1, C i = P i+1 (for i = 1,., N 1), C N = Decision Variable t= Length of single operation period (time interval) SL c = Service level at the customer shop n Tc = Number of trucks transporting products between the customer shop and the 1 st supplier SL i = Service level at the warehouse of the i th supplier n Ti = Number of the trucks transporting products between the i th supplier and the (i + 1) th supplier n Mi = Number of machines at the i th supplier K i = Number of production Kanbans at the i th supplier 3.4 Other Notation i= Index of the suppliers ω= Random variable denoting the number of withdrawal Kanbans that are collected at the customer shop per time interval K ωc = Number of withdrawal Kanbans at the customer shop b c = Approximate distribution of the number of withdrawal Kanbans to be carried to the 1 st supplier per time interval n TUc = Random variable denoting the number of serviceable trucks transporting products between the customer shop and the 1 st supplier at the beginning of each time interval b ac = Distribution of the number of withdrawal Kanbans mange to arrive at the 1 st supplier per time interval S c = Randon variable denoting the amount of satisfied demand at the customer shop per time interval K ωi = Number of withdrawal Kanbans in the raw material warehouse of the i th supplier n MUi = Random variable denoting the number of serviceable machines at the i th supplier at the beginning of each time interval 23

37 b i = Distribution of the number of withdrawal Kanbans to be carried from the i th supplier to the (i + 1) th supplier per time interval n TUi = Random variable denoting the number of serviceable trucks at the i th supplier at the beginning of each time interval (i = 1,2,, N 1) b ai = Distribution of the number of withdrawal Kanbans mange to arrive at the (i + 1) th supplier per time interval n i = Random variable denoting the number of finished products in the product output buffer of the i th supplier S i = Random variable denoting the amount of satisfied demand at the i th supplier per time interval 3.5 Objective Function Total profit generated from the supply chain = GrossProfit DemandRejectionCost InventoryCost EquipmentCost N i=1 = 1 {(P t c C c )E(S c ) + (P i C i )E(S i )} {R c [Dt E(S c )] + R 1 [E(S c ) N i=2 N 1 thi i [2K ωi E(S i )] 2 N E(S 1 )] + R i [E(S i 1 ) E(S si )]} { (2K ωc Dt)tH c 2 N 1 N + i=1 tho i E(n i ) + i=1 } {(T c tn Tc + G c n Tc ) + i=1 (T i tn Ti + G i n Ti ) + i=1 M i tn Mi } 3.6 Explanation Gross Profit The total gross profit made by the entire supply chain per operational interval is denoted as follows, (P c C c )E(S c ) + (P i C i )E(S i ) E(S c ) is the long-run average amount of demand that will be satisfied at the customer shop per operational interval. (P c C c ) is the unit gross profit of satisfied demand at the customer shop. (P c C c )E(S c ) is the expected total gross profit made at 24 N i=1

38 the customer shop per operational interval. Similarly, E(S i ) is the long-run average amount of demand that will be satisfied at the i th supplier per operational interval. (P i C i ) is the unit gross profit of satisfied demand at the i th N supplier. And i=1 (P i C i )E(S i ) is the expected total gross profit made by all the suppliers per operational interval. (P c C c ) and (P i C i ) are just financial parameters determined by the market of a specific industry. So we will focus on explaining the derivations of E(S c ) and E(S i ). In order to answer the question above, we need to start from the beginning of the supply chain --- initial customer demand. We assume that the initial customer demand follows a Poisson distribution with a mean value Dt, where D is the average demand that arrive at the customer shop per unit time, and t is the length of the operational interval of the synchronized supply chain. We use random variable ω to denote the number of customer demands that arrive at the customer shop per operational interval. Therefore, P(ω = ω t = t) = (Dt)ω ω! e Dt, ω = 0,1,2 (1) In our model, the service level at the customer shop (SL c ) is a predetermined parameter. By setting up the preferred service level at the customer shop, we can calculate the optimum number of withdrawal Kanbans at the customer shop with the following equation, K ωc = the smallest K ω that satisfies P(ω = ω t = t) SL c (2) In the customer shop, when the products are bought by customers, the withdrawal Kanbans are detached from these products and collected in a Kanban-post. The customer shop then requests products from the 1 st supplier in regular operational interval t by carrying the collected withdrawal Kanbans to the 1 st supplier s output inventory as retrieving information. The requested products are taken from the finished-goodinventory of the 1 st supplier s plant and attached with these incoming withdrawal 25 K ω ω=0

39 Kanbans, and are then carried back to the customer shop to replenish its inventory for future customer. The distribution of the number of withdrawal Kanbans collected in the Kanban-post during each operational interval can be approximated as: b c = ( b c0 b c1 b c(kωc 1) b ckωc ) = P(ω = 0 t = t) P(ω = 1 t = t) P(ω = K ωc 1 t = t) K ωc 1 1 P(ω = ω t = t) ( ω=0 ) Note that the equation above is based on a strong assumption that all the (3) withdrawal Kanbans sent to the supplier must have been taken back to the customer shop before the beginning of each new operational period. However, it is not always the case in practice. For example, if the requested products are not available at the supplier s output buffer in the current time interval, then withdrawal Kanbans have to wait there until the next time period. Therefore, the realistic service level at the customer shop will not always be equal to the predefined service level. However, this kind of simplification is acceptable in modelling. The withdrawal Kanbans collected in the Kanban-post are to be carried to the supplier by trucks. Thus, we need to take into account the transportation risk between the customer shop and 1 st supplier, caused by the failure of the trucks. We assume that the trucks could only be detected down before the beginning of each round trip. As long as a truck manages to leave the customer shop, it will always come back before the beginning of the next operation interval. Some temporary denotations are as follows, UD t = probility that a truck changes from up to down in one single time interval UU t = probility that a truck remains up in one single time interval 26

40 DU t = probility that a truck changes from down to up in one single time interval DD t = probility that a truck remains down in one single time interval Then, UD t = 1 e tλ Tc UU t = 1 UD t = e tλ Tc DU t = 1 e t β Tc DD t = 1 DU t = e t β Tc We use binary digits to denote the states of each truck, 1 means the truck is up and 0 means the truck is down. So the transition matrix of each single truck is as follows, [ UU t UD t DU t DD t ] Following the same pattern, we can build the Markov transition matrix of all the available trucks by comparing the binary vector of each states and multiplying each corresponding probabilities. For example, if we have 2 trucks, the total transition matrix will be like follows, DD t DD t DD t DU t DU t DD t DU t DU t 01 DD [ t UD t DD t UU t DU t UD t DD t UU t ] 10 UD t DD t UD t DU t UU t DD t UU t DU t 11 UD t UD t UD t UU t UU t UD t UU t UU t Based the transition matrix, we can then calculate the long-run distribution of the number of working trucks at the beginning of each round trip. For example, if we have 2 trucks, P(n Tcu = 1) = π(01) + π(10). P(n TUc = 0) P(n TUc = 1) =Stationary probabilities of the Markov transition matrix (4) P(n TUc = n Tc 1) ( P(n TUc = n Tc ) ) 27

41 Now, we have the distribution of the number of withdrawal Kanbans collected in the Kanban-post and the distribution of the number of available trucks at the beginning of each time interval. By combining the information from these two distributions, along with the transportation capacity of each truck, we can get the distribution of the number of withdrawal Kanbans that actually arrive at the upstream supplier in each single time period. The calculation is shown as follows. b ac = ( b ac0 b ac1 b ac(kωc 1) b ackωc ) where b ack = b cx P(n TUc = y), k = min(x, yc Tc ) for { x = 0,1,2, K ωc y = 0,1,2, n Tc The logic of equation (5) is simple. We consider every combination of the elements from the two right-hand distributions and decide the number of withdrawal Kanbans that can actually be transported under that combinational situation. More specifically, we compare the total transportation capacity and total transportation demand under each combinational situation, and take the minimum value as the actual arriving number. We record the corresponding joint probability of each combination and sum up the joint probabilities whose actual arriving number are the same. Finally, we put the joint probabilities sums into the left-hand distribution to finish the calculation. In the long run, the average amount of demand that will be satisfied at the customer shop is equal to the average amount of withdrawal Kanbans actually carried to the 1 st supplier during each time interval. The reason is that if the customer demand is much bigger than the transportation capacity then the actually fulfilled demand will be restrained by the transportation capacity; On the other hand, if the customer demand is much smaller than the transportation capacity, then the distribution of the customer demand will not be distorted by the transportation capacity. Therefore, 28 (5)

42 K ωc E(S c ) kb ack k=0 (6) We organize all the elements of b ac in a specific way to obtain Matrix S 1, as shown in equation (7). Generally, Matrix S denotes the contribution of the demand from a downstream facility to the change of state of its upstream supplier s plant. More detailed explanation of the S matrix is provided by Matzka et al. (2012). S 1 = b ac0 b ac1 b ac2 b ackωc b ac0 b ac b ac b ac0 b ac1 1 b acx b ac0 1 b ac0 [ ] (z 1 + 1) (z 1 + 1) where, z 1 = K ωc + K 1. Note that K 1 has its impact on the size of Matrix S 1. 1 x=0 (7) Using the similar way in which we construct the distribution of the number of working trucks, we can calculate the distribution of the number of working machines in the supplier s plant. P(n MU1 = 0) P(n MU1 = 1) =Stationary probabilities of the Markov transition matrix ( P(n MU1 = n M1 1) P(n MU1 = n M1 ) ) (8) Where the transition matrix is constructed with n M1, λ M1, β M1 and t, based on the assumption that the both of a machine s failure and repairing processes follow exponential distribution. 29

43 As the counterpart of matrix S, matrix Q denotes the contribution of manufacturing section of the supplier s plant to the change of the state of this plant. The Q 1 matrix, for example, can be constructed as shown in equation (9), Q 1 = [q 1xz ] for { Where, x = 0,1,2,, z 1 y = 0,1,2,, n M1, z = 0,1,2,, z 1 (z 1 + 1) (z 1 + 1) q 1xz = { P(n M1u = y), {x + min(x, ytc M1 )} = z 0, else In order to decide how many parts could be produced during each time interval in a supplier s plant, two factors need to be taken into account --- the available production Kanbans circulating in the manufacturing process and the total production capacity. The production Kanbans circulate between the finished products inventory and the plant. Each finished product in the supplier s output inventory is attached with a production Kanban. When a new unit of demand arrives from the downstream facility, the attached production Kanban will be detached from the finished product which is to be carried away, and that detached production Kanban will enter the manufacturing process. When a new unit product is finished, the production Kanban will be again attached to that product and moved to the output inventory. According to the pulling policy, the manufacturing section can only start to process a new product when it gets a new production Kanban. Production capacity is another issue that needs to be concerned. If the number of production Kanbans exceeds the total production capacity during a specific operational interval, only the part of production demand that is under the capacity would be manufactured. More detailed explanation about how to decide the number of production (9) 30

44 Kanbans in the manufacturing section given a specific state of the supplier s plant can be referred to in Matzka et al. (2012) s paper. Therefore, the actual production amount in each time interval is the minimum value between the number of production Kanbans in the manufacturing section and the total production capacity. Recall that matrix Q denotes the contribution of manufacturing section of the supplier s plant to the change of the state of this plant, and Matrix S denotes the contribution of the demand from the downstream facility to the change of state of its upstream supplier s plant. Therefore, if we times matrix Q by matrix S, we will get a matrix that denote the state of the supplier s plant which contains the factors of both downstream demand side and upstream supply side. And we call this new matrix P in general. As for the 1 st supplier, we calculate P 1 as follows, of P 1, and P 1 = Q 1 S 1 (10) x 1 = (x 10, x 11,, x 1z1 ) is the row-vector denoting the stationary probabilities z 1 P(n 1 = 0) = x 1y y=k 1 P(n 1 = 1) = x 1K1 1 P(n 1 = 2) = x 1K1 2 P(n 1 = K 1 1) = x 1[K1 (K 1 1)] = x 11 { P(n 1 = K 1 ) = x 1(K1 K 1 ) = x 10 In equation (11), we calculated the distribution of the number of finished products in the output inventory of the 1 st supplier s plant. More detailed explanation about how to decide the number of finished products in the inventory of a supplier s plant given a specific state of the supplier s plant is available in Matzka et al. (2012) s paper. And the (11) 31

45 expected number of finished products in the output inventory of the 1 st supplier can be calculated as follows, and we will talk about the use of E(n 1 ) later. K 1 E(n 1 ) = xp(n 1 = x) x=0 By considering each combination of the elements from the distribution of the (12) number of withdrawal Kanbans arriving at the 1 st supplier, and the distribution of the number of finished products in the inventory of the 1 st supplier s plant, we can get the distribution of actual satisfied demand, as shown in equation (13). x = 0,1,2,, K 1 P(S 1 = z) = P(n 1 = x)b acy, z = min (x, y) for { y = 0,1,2,..., K ωc (13) z = 0,1,2,..., K ωc The logic behind equation (13) is pretty straightforward, we simply pick the minimum value between the number of incoming demands, and the number of available products on hands. And the expected value of the number of satisfied demand at the 1 st supplier s plant can be calculated as follows, K ωc E(S 1 ) z P(S 1 = z) (14) z=0 Recall the expression of the total gross profit (P c C c )E(S c ) + N i=1 (P i C i )E(S i ), we can get E(S c ) with equation (6), and E(S 1 ) with equation (15). And the logic to get E(S i ) for i = 2,3,, N is very similar. N Now, we fully understand the derivation of (P c C c )E(S c ) + i=1 (P i C i )E(S i ), which is the total of the gross profits from each facility in the supply chain. For each facility, its gross profit value is equal to the expected amount of satisfied demand multiplies the unit gross profit of its product. 32

46 3.6.2 Demand Rejection Cost In the age of e-commerce, customer rejection cost can be considerable. There are so many B2C websites, which provides very similar products to their customers. If your delivery speed or inventory level cannot fulfill your customers fussy requirement, they could just click away and go to your competitor s website. Therefore, it is essential to include demand rejection cost in our objective function. In our model, the expression of demand rejection cost is as follows, R c [Dt E(S c )] + R 1 [E(S c ) E(S 1 )] + R i [E(S i 1 ) E(S si )] We consider each facility as an entity with two interfaces --- one interface for inbound demand and one interface for outbound demand. And the outbound demand for a downstream entity is the inbound demand for its upstream entity. This relationship is illustrated in Figure 3-2. N i=2 Figure 3-2 Demand flow Take the customer shop and the 1st supplier for example, their relationship is shown in Figure 3-3. Figure 3-3 Demand flows through customer shop and 1st supplier The inbound demand of the customer shop is the initial customer demand Dt. The outbound demand of the customer shop is the proportion of initial customer demand that 33

47 is actually satisfied by the customer shop E(S c ). According to the pulling philosophy, the customer inventory that is used to satisfy initial customer demand needs to be replenished by withdraw the parts from the 1 st supplier. Therefore, E(S c ) is also the inbound demand of the 1 st supplier. And the outbound demand of the 1 st supplier is E(S 1 ), which is the proportion of E(S c ) that is actually satisfied by the 1 st supplier. Meanwhile, E(S 1 ) is also the inbound demand of the 2 nd supplier. So on and so forth. For the customer shop, [Dt E(S c )] is the proportion of initial customer demand that is not satisfied, or rejected, by the customer shop. R c is the unit demand rejection cost at the customer shop. Therefore, R c [Dt E(S c )] is the total expected demand rejection cost at the customer shop during each time interval. For the 1 st supplier, [E(S c ) E(S 1 )] is the proportion of the demand from customer shop that is not satisfied, or rejected, by the 1 st supplier. R 1 is the unit demand rejection cost at the 1 st supplier. Therefore, R 1 [E(S c ) E(S 1 )] is the total expected demand rejection cost at the 1 st supplier during each time interval. Similarly, for the i th supplier (i = 2,3,, N), [E(S i 1 ) E(S i )] is the proportion of demand from the (i 1) th supplier that is not satisfied, or rejected, by the i th supplier. R i is the unit demand rejection cost at the i th supplier. Therefore, R i [E(S i 1 ) E(S i )] is the total expected demand rejection cost at the i th supplier during each time interval. N Therefore, R c [Dt E(S c )] + R 1 [E(S c ) E(S 1 )] + i=2 R i [E(S i 1 ) E(S si )] is the total customer rejection cost from all the facilities of the supply chain Inventory Cost The total inventory cost of the supply chain is (2K ωc Dt)tH c 2 N + tho i E(n i ) + thi i [2K ωi E(S i )] 2 i=1 34 N 1 i=1

48 The inventory cost at the customer shop is (2K ωc Dt)tH c 2.As shown in the Figure 3-4, at the beginning of each operation period, there are K ωc amount of products stock in the inventory of the customer shop, each of which is attached with a withdrawal Kanban. On average, there will be Dt amount of products taken away by the customers during each time interval. Therefore, at the end of each operation interval, there are (K ωc Dt) Inv c (t) K ωc K ωc Dt 0 t Figure 3-4 Inventory plot of the customer shop t amount of products left in customer shop s inventory. To simply our calculation, we assume that the consumption rate of the products is a constant. Therefore, The inventory cost at the customer shop = H c 0 t Inv c (t)dt (2K ωc Dt)tH c 2 As for each supplier, there are two kinds of inventory. First is the output inventory in which the finished products, manufactured by the supplier s plant, are stocked. Second is the raw material inventory of the supplier s warehouse, which supplies the raw material to the supplier s plant. Therefore, there are also two sorts of inventory cost for each supplier --- the output inventory cost and the raw material inventory cost (input inventory cost). 35

49 The output inventory cost of the i th supplier is tho i E(n i ). In equation (12), we get the value of E(n 1 ), which is the expected number of finished products in output inventory of the 1 st supplier. Therefore, tho 1 E(n 1 ) is the expected total output inventory cost of the 1 st supplier during each time interval. For the succeeding suppliers. The calculations are very similar. And we will talk about them later. In summary, N i=1 tho i E(n i ) is the expected total output inventory cost of all the suppliers in the supply chain. In order to talk about each supplier s raw material inventory cost, we need to continue to elaborate our model. Mathematically, the raw material warehouse functions similarly like the customer shop. Take the 1st supplier s warehouse for example. The inbound demand of the 1st supplier s raw warehouse comes from the consumption of the raw material at the same supplier s plant, which is S 1. K ω1 = the smallest K ω that satisfies P (S 1 = S 1 ) SL 1 (15) When one unit of raw material is to be taken away to the manufacturing section, its 36 K ω S 1 =0 The service level at the raw material warehouse of the 1st supplier (SL 1 ) is also a variable to our model. By setting up the preferred service level at the 1st supplier s warehouse, we can calculate the optimum number of withdrawal Kanbans at the raw material warehouse with equation (15). For each supplier, its plant and raw material warehouse are close to each other. So we assume no communication barrier between the plant and the warehouse. Therefore, no Kanbans are needed between the plant and the warehouse. The plant takes raw material from the warehouse in the same way the customer takes final products from the customer shop. However, the raw material in the warehouse are attached with withdrawal Kanbans.

50 withdrawal Kanban will be detached and collected in a Kanban-post. The warehouse of 1st supplier then requests products from the 2nd supplier in regular operation period t by carrying the collected withdrawal Kanbans to the 2 nd supplier as retrieving information. The requested products are taken from the output inventory of the 2 nd supplier and attached with these incoming withdrawal Kanbans, and are then carried back to the warehouse of the 1st supplier to replenish its raw material inventory for future production. Now we can explain the supplier s raw material inventory cost. The warehouse inventory cost of the i th supplier is expressed as N 1 thi i [2K ωi E(S i )] i=1 2. In the case of the warehouse of the 1 st supplier, it is thi 1 [2K ω1 E(S 1 )]. As shown in Figure 3-5, at the 2 beginning of each operation interval, there are K ω1 amount of raw material stocked in the warehouse of the 1 st supplier, each of which is attached with a withdrawal Kanban. On average, there will be E(S 1 ) amount of raw material taken away by plant of the same supplier during each time interval. Therefore, at the end of each operation interval, there are K ω1 E(S 1 ) amount of raw material left in warehouse of the 1st supplier. To simply our calculation, we assume that the consumption rate of the raw material is a constant. Inv S1 (t) K ω1 K ω1 E(S 1 ) 0 t t Figure 3-5 Inventory plot of the 1st supplier s warehouse 37

51 Therefore, Inventory cost of 1 st supplier s warehouse = HI 1 0 t Inv S1 (t)dt thi 1[2K ω1 E(S 1 )] 2 The same method is applicable for all the other suppliers warehouse in the supply N 1 chain. Therefore, i=1 is the total expected raw material warehouse inventory thi i [2K ωi E(S i )] 2 cost for all the suppliers in the supply chain. Note that there s no raw material warehouse in the most upstream supplier because we assume the resource site at the last supplier provides infinite raw material. In summary, (2K ωc Dt)tH c 2 N + i=1 tho i E(n i ) + inventory cost of all the facilities in the supply chain Equipment Cost N 1 thi i [2K ωi E(S i )] 2 i=1 is the total expected follows, The total cost of running all the equipment of the supply chain is denoted as N 1 N (T c tn Tc + G c n Tc ) + (T i tn Ti + G i n Ti ) + M i tn Mi i=1 i=1 There are two kinds of equipment in the supply chain --- the trucks carrying withdrawal Kanbans and products between the facilities, and the machines operating in each supplier s plant. T c tn Tc is the total depreciation cost of all the trucks travelling between the customer shop and the 1 st supplier s plant. T i tn Ti is the total depreciation cost of all the trucks travelling between the i th supplier s raw material warehouse and the (i + 1) th supplier s plant. Therefore, T c tn Tc + N 1 i=1 T i tn Ti is the total depreciation cost of all the trucks in the supply chain. Note that we assume the last supplier is located at its raw 38

52 material source so that no truck is needed between the last supplier and its material source. G c n Tc is the total gasoline cost of all the trucks travelling between the customer N 1 shop and the 1st supplier s plant. And i=1 G i n Ti is the total gasoline cost of all the trucks travelling between the i th supplier s raw material warehouse and the (i + 1) th N 1 supplier s plant. Therefore, G c n Tc + i=1 G i n Ti is the total gasoline cost of all the trucks in the supply chain. M i tn Mi is total depreciation cost of all the machines in the i th supplier s plant. N Therefore, i=1 M i tn Mi is the overall depreciation cost of all the machines in the supply chain. N 1 In summary, (T c tn Tc + G c n Tc ) + i=1 (T i tn Ti + G i n Ti ) + i=1 M i tn Mi is the overall equipment running cost of the supply chain Net Profit N period as: NetProfit Next, we can calculate the net profit made by this supply chain in each operation = GrossProfit DemandRejectionCost InventoryCost EquipmentDepreciationCost N = {(P c C c )E(S c ) + i=1 (P i C i )E(S i )} {R c [Dt E(S c )] + R 1 [E(S c ) E(S 1 )] + N R i [E(S i 1 ) E(S si )]} { (2K ωc Dt)tH c N + tho i E(n i ) i=2 N 1 thi i [2K ωi E(S i )] 2 2 i=1 + i=1 } {(T c tn Tc + G c n Tc ) + i=1 (T i tn Ti + G i n Ti ) + M i tn Mi Unit Time Measurement N 1 N i=1 } So far, we have obtained the net profit made in each time interval. In the next chapter, we will search for the optimum solution to reach the maximum value of the 39

53 objective function. And the length of the operation interval is one of the variables in the solution space. Therefore, comparing the net profit of each time interval is not a fair way to evaluate each feasible solution. For example, when we set the time interval to be 2 hours, the net profit of each time interval is, say, $1000, when we set the time interval to be 0.5 hour, the net profit of each time interval is, say, $300. Even though 1000 >300, (1000/2) < (300/0.5). Therefore, we had better compare the net profit made by the supply chain in each unit time, rather than in each operational interval. After this modification, we get the final form of our objective function: Total profit generated from the supply chain = NetProfit = 1 {(P t c N C c )E(S c ) + i=1 (P i C i )E(S i )} {R c [Dt E(S c )] + R 1 [E(S c ) E(S 1 )] + N i=2 R i [E(S i 1 ) E(S si )]} { (2K ωc Dt)tH c N 1 thi i [2K ωi E(S i )] 2 2 N + i=1 tho i E(n i ) + i=1 } {(T c tn Tc + G c n Tc ) + i=1 (T i tn Ti + G i n Ti ) + i=1 M i tn Mi } 3.7 Iteration In equation (15), we get the maximum number of withdrawal Kanbans in the raw material warehouse of the 1st supplier. Equation (16) provides the distribution of the number of withdrawal Kanbans N 1 collected in the Kanban-post in the warehouse of the 1st supplier. b 1 = ( b 10 b 11 b 1(Kω1 1) b 1Kω1 ) = P(S 1 = 0) P(S 1 = 1) P(S 1 = K ω1 1) K ω1 1 1 P(S 1 = S 1 ) ( S 1 =0 ) Similar to equation (4), based on the value of n T1, λ T1, β T1 and t, we can calculate the number of working trucks at the 1 st supplier s warehouse at the beginning of each t N (16) 40

54 time interval with equation (17). Note that these trucks travel between the warehouse of the 1st supplier and the 2 nd supplier. P(n TU1 = 0) P(n TU1 = 1) P(n TU1 = n T1 1) =Stationary probabilities of the Markov transition matrix (17) ( P(n TU1 = n T1 ) ) Similar to equation (5), we can calculate the distribution of the number of withdrawal Kanbans actually carried to 2nd supplier s output buffer during each operational interval, with equation (18). b a1 = ( b a10 b a11 b a1(kωc 1) b a1kωc ) where b a1k = b a1x P(n TU1 = y), k = min(x, yc T1 ) for { x = 0,1,2, K ω1 y = 0,1,2, n T1 Here begins the iterations of all the calculations for the rest of the suppliers. And i=2,3, N for all the equations below. For each iteration, we use equation (19) to generate the S matrix for the i th supplier. (18) S i = b a(i 1)0 b a(i 1)1 b a(i 1)2 b a(i 1)Kω(i 1) b a(i 1)0 b a(i 1) b a(i 1) b a(i 1)0 b a(i 1)1 1 b a(i 1)x b a(i 1)0 1 b a(i 1)0 [ ] (z i + 1) (z i + 1) where z i = K ω(i 1) + K i. 1 x=0 (19) For each iteration, based on the value of n Mi, λ Mi, β Mi and t, we use equation (20) to generate the distribution of the number of working machines in the i th supplier s plant at the beginning of each time interval. 41

55 P(n MUi = 0) P(n MUi = 1) P(n MUi = n Mi 1) =Stationary probabilities of the Markov transition matrix (20) ( P(n MUi = n Mi ) ) supplier. For each iteration, we use equation (21) to generate the Q matrix for the i th Q i = [q ixz ] for { Where, x = 0,1,2,, z i y = 0,1,2,, n Mi, z = 0,1,2,, z i q ixz = { P(n MUi = y), {x + min(x, ytc Mi )} = z 0, else (21) For each iteration, we use equation (22) to obtain the P matrix for the i th supplier. P i = Q i S i (22) x i = (x i0, x i1,, x izi ) is the Row-vector denoting the stationary probabilities of P i, then we can calculate the distribution of the number of finished products stocked in the i th supplier s output inventory, with equation (23). z i P(n i = 0) = x iy y=k i P(n i = 1) = x iki 1 P(n i = 2) = x iki 2 P(n i = K i 1) = x i[ki (K i 1)] = x i1 { P(n i = K i ) = x i(ki K i ) = x i0 For each iteration, we use equation (24) to calculate the expected inventory level in the output inventory of the i th supplier s plant. (23) K i E(n i ) = xp(n i = x) x=0 (24) 42

56 For each iteration, we use equation (25) to calculate the distribution of the amount of demand that are satisfied by the i th supplier. x = 0,1,2,3,, K i P(S i = z) = P(n i = x)b a(i 1)y, z = min (x, y) for { y = 0,1,2,..., K ω(i 1) (25) z = 0,1,2,..., K ω(i 1) For each iteration, we use equation (26) to calculate the expected amount of demand that are satisfied by the i th supplier. K ω(i 1) E(S i ) z P(S i = z) (26) z=0 For each iteration, we use equation (27) to calculate the optimum number of withdrawal Kanbans in the raw material warehouse of the i th supplier. K ωi = the smallest K ω that satisfies P (S i = S i ) SL i (27) For each iteration, we use equation (28) to calculate the distribution of the number of withdrawal Kanbans that are collected in the Kanban-post of the warehouse of the i th supplier. K ω S i =0 b i = ( b i0 b i1 b i(kωi 1) b ikωi ) = P(S i = 0) P(S i = 1) P(S i = K ωi 1) K ωi 1 1 P(S i = S i ) ( S i =0 ) (28) Based on the value of n Ti, λ Ti, β Ti and t, we use equation (29) to calculate the distribution of the number of working trucks at the warehouse of the i th supplier at the beginning of each operational interval. 43

57 P(n TUi = 0) P(n TUi = 1) P(n TUi = n Ti 1) = Stationary probabilities of the Markov transition matrix (29) ( P(n TUi = n Ti ) ) For each iteration, we use equation (30) to calculate the amount of withdrawal Kanbans that are actually carried from the i th supplier s warehouse to the (i + 1) th supplier during each time interval. b ai = ( b ai0 b ai1 b ai(kωi 1) b aikωi ) where b aik = b aix P(n TUi = y), k = min(x, yc Ti ) for { x = 0,1,2, K ωi y = 0,1,2, n Ti Then, we iterate (19)-(30) till i = N. (30) When i = N, we only repeat (19)-(26), because there s no upstream facility for the last supplier 3.8 Constraint The constraints of our model have already been implicitly expressed in the equations. For example, according to equation 19, the number of possible states of a supplier s material outbound interface is always constrained by the sum of the number of withdrawal kanbans from its downstream supplier s information outbound interface and the number of production kanbans at this supplier s plant; according to equation (21), the throughput of a plant during a single period is always constrained by minimum of the number of working machines and the number of production Kanban released in that plant; according to equation (30), the inbound demand for a supplier is always constrained to the minimum of the number of collected withdrawal Kanban at its downstream supplier and this downstream supplier s transportation capacity. 44

58 4 SOLUTION METHODOLOGY Simulated Annealing is a meta-heuristic inspired by the physical annealing process studied in statistical mechanics (Aarts et al., 1988). SA sometimes allows to accept worse solutions in order to escape from local optimum, and the general idea is like follows. Within the domain of our decision variables (the search space of our model), we move from an old point to a new point. The old point corresponds to an old objective value, and the new point corresponds to a new objective value. Then, for a maximization problem, objectiveimprovement we compare e temperature with a number randomly generated between zero and objectiveimprovement one. If e temperature is bigger, we accept the new point as our current optimum solution and the new objective value as our current maximum objective value. We continuously move from old points to new points until a desired objective value is obtained (Goffe et al., 1994). When objectiveimprovement is positive, which means the new objective value objectiveimprovement is bigger, e temperature is always bigger than one, so the new point will always be accepted, as it should be. When objectiveimprovement is negative, which means the objectiveimprovement new objective value is smaller, e temperature still has a chance to be bigger than the random number. In this way, some worse solutions will be accepted. Note that, when objectiveimprovement is negative, the bigger the objectiveimprovement, the higher the chance that the new point will be accepted, which means the new point is not that worse anyway (Kirkpatrick et al., Bouleimen et al., 2003). Also, temperature is always a positive number. Therefore, the higher the temperature, the higher the chance that the new point will be accepted. As the SA goes, temperature slowly decreases, so the chance that a worse solution will be accepted is getter smaller and smaller. When temperature finally reaches zero, no worse solution 45

59 will be accepted anymore. In this way, the current maximum objective value slowly converges to the final maximum objective value (Van et al., Bouleimen et al., 2003). However, in order to actually implement SA, we need to know more details about the key components of it so that we can apply it to solving our own problem. 4.1 Solution Vector Recall the decision variable section in chapter 3, if we put all of our variables in a row then we get our solution vector: x = [t, SL c, n Tc, SL 1 SL N 1, n T1 n T(N 1), n M1 n MN, K 1 K N ] 4.2 Objective Function The objective function of our Kanban model describes the total profit generated through supply chain. Recall section 3.4, the initial formulation of P SC consists of decisions variables, parameters and intermediate notations. The parameters are merely constant values, and the intermediate notations are all mapped from the decision variables. Therefore, the entire objective function is mapped from the decision variables, which means we can abstract it as f(x) = Total profit generated from the supply chain Later on, we will embed f(x) into our SA algorithm. 4.3 Search Space The search space of our SA is S(x). For the sake of efficiency, we want S(x) to be narrow enough so that we only focus on the most high quality range of each element in x. In order to decide the high quality range for a specific decision variable x, we need to hold any other variable at a reasonable level, given a predetermined set of parameters. 46

60 Then we can run some tests to find out the relationship between f(x) and x so that the high quality range of x could be discovered. After figuring out these ranges of all the elements in x, the combination of these ranges become our S(x). 4.4 Perturbation Perturbation is the process of obtaining the next solution vector x of the current solution vector x within S(x). Shahabudeen et al. (2002) mentioned two perturbation algorithms, and we modify these algorithms to make them fit our specific problem Perturbation Algorithm One Step 1: Consider the current solution vector [t, SL c, n Tc, SL 1 SL N 1, n T1 n T(N 1), n M1 n MN, K 1 K N ] Step 2: Set a tag for each of the above elements with equal probability Step 3: Generate a uniform random number r 1 Step 4: Based on r 1 select the element to be changed Step 5: Generate a random number r 2 If (r 2 < 0.5) Increment the element subject to its upper limit Else Decrement the element subject to its lower limit Step 6: Set the new solution vector Perturbation Algorithm Two Step 1: Consider the current solution vector [t, SL c, n Tc, SL 1 SL N 1, n T1 n T(N 1), n M1 n MN, K 1 K N ] Step 2: Do the following for each of the elements 47

61 2.1 Generate a random number r 2.2 Based on r perform any one of the following with equal probability for the parameters: (a) Increment the value subject to its upper limit (b) Decrement the value subject to its lower limit (c) No change in the value Step 3: Set the new solution vector 4.5 Cooling Schedule As stated before, a SA iterates for several times at a particular temperature. In our study, the number of iterations at a temperature level is not fixed. Instead, we use ACCETP and TOTAL together to decide it dynamically. ACCEPT counts the number of accepted solutions at a particular temperature and TOTAL counts the number of total generated solutions at the same temperature. We stop the iterations when its ACCEPT reaches the accept limit (beta) or its TOTAL reaches four times beta. When the iterations at a current temperature stop and the stoppings rules have not been met, we multiply the current temperature by a cooling factor (alpha) to obtain the next temperature level. 4.6 Stopping Rules Our SA stops when it meets either of the following two criteria: (1) The temperature drops below the final temperature. (2) The freeze count reaches its predetermined threshold (pi).we calculate the freeze count after the iterations stop at a particular temperature. If ACCEPT TOTAL 0.15, we increment the freeze count by one. Otherwise, it stays the same. Whenever a better solution is found, we reset the freeze count to zero. 48

62 4.7 SA Pseudocode Now that we have introduced all the components, let s use pseudocode to demonstrate how they function all together under a SA framework SA parameters t i = initial temperature t f = final temperature α = cooling factor β = accept limit π = freeze limit X 0 = initial solution vector SA variables f m = maximum objective value temp = current temperature X m = best solution vector acc = # accepted solution X c = current solution vector tot = # generated solution r = random number between zero and one c f = freeze count SA pseudocode acc = 0; tot = 0; c f = 0; temp = t i ; X c = X 0 ; f m = f(x c ); X m = X c ; while temp > t f and c f < π X c = Perturbation(X c ); if f(x c ) > f m else f m = f(x c ); X m = X c ; c f = 0; acc = acc + 1; tot = tot + 1; if acc > β or tot > 4β temp = temp α; If acc tot 0.15 c f = c f + 1; acc = 0; tot = 0; f(xn ) f m if e temp > r f m = f(x n ); X m = X c ; c f = 0; acc = acc + 1; tot = tot + 1; if acc > β or tot > 4β 49

63 else tot = tot + 1; temp = temp α; if acc tot 0.15 c f = c f + 1; acc = 0; tot = 0; if acc > β or tot > 4β temp = temp α; if acc tot 0.15 c f = c f + 1; acc = 0; tot = 0; 50

64 5 RESULT In this chapter, we will first implement SA to optimize two Kanban-based supply chains. One is a one-customer-shop & one-supplier (C-S) supply chain, and the other one is a one-customer-shop & two-supplier (C-S-S) supply chain. Then, we will take one step further to seek for deeper insights into the Kanban system performance through model analysis, based on our optimum configuration of the C-S-S supply chain. 5.1 Kanban Optimization C-S Supply Chain First of all, we assign some numerical values to the parameters of this C-S supply chain, as shown in Figure 5-2. Based on these parameters, we are able to map a specific solution vector to its objective value. Figure 5-1 One-customer-shop and one-supplier (C-S) supply chain 51

65 Table 5-1 Parameters of the C-S supply chain Customer Shop Supplier System Parameter D c Tc λ Tc β Tc c M1 λ M1 β M Financial Parameter P C C c R c H c T c G c P 1 C 1 R 1 HO 1 M According to our model, there are five decision variables for a C-S supply chain, and they are the length of one single operation period, the service level at the customer shop, the number of trucks travelling between the customer shop and the supplier, the number of production Kanbans in the supplier s plant and the number of machines in the supplier s plant. Therefore, the solution vector of a C-S supply chain can be expressed as x 1 = [t, SL c, n Tc, K 1, n M1 ]. If we abstract our Kanban model to a function f, then the objective value of a C-S supply chain is f(x 1 ) = f[t, SL c, n Tc, K 1, n M1 ]. And our goal is find the optimum x 1 and its corresponding maximum f(x 1 ). Next, we need to decide the search space. In other words, we want to figure out the search range for each of the five variables in x 1. K 1 is the first variable we want to focus on. In order to discover the relationship between K 1 and f( K 1 ), the other four variables need to stay constant while K 1 varies. We set t = 1 for the simplicity of calculation, SL c = 80% because of the 80/20 principle, n Tc = 4 because the single truck capacity is 3 so that 4 trucks should cover the average demand rate of 10, and n M1 = 4 for the same reason as n Tc. According to Table 5-2, f( K 1 ) begins to increase dramatically when K 1 = 10 and decrease when K 1 = 30. Therefore, we set [10, 30] to be the search range for K 1, and 2 to be its search step. Table 5-2 K 1 experiment on the C-S supply chain t SL c n Tc n M1 K 1 f( K 1 ) 1 80% % %

66 1 80% % % % Now we move to variable SL c. In Table 5-2, f( K 1 ) reach its maximum value when K 1 = 25. Therefore, we set K 1 = 25 and relax SL c. According to Table 5-3, f(sl c ) begins to increase when SL c = 65% and decrease when SL c = 85%. Hence, we set [65%, 85%] to be the search range for SL c, and 2% as its search step. Table 5-3 shows that f(sl c ) reaches a local maximum when SL c = 75%. Therefore, we set SL c = 75% and relax our next variable t. According to Table 5-4, f(t) begins to increase when t = 0.5 and decrease when t = 1.5. Therefore, we set [0.5, 1.5] to be the search range of t, and 0.1 as its search step. Also, f(t) reaches its local maximum when t = 1. Table 5-3 SL c experiment on the C-S supply chain t n Tc K 1 n M1 SL c f(sl c ) % % % % % % % % Table 5-4 t experiment on the C-S supply chain SL c n Tc K 1 n M1 t f(t) 75% % % % % %

67 As for variables n Tc and n M1, pilot study shows a safe margin of +1 should guarantee their high performance search range. Therefore, we set [3, 5] to be the search range for both n Tc and n M1, and 1 to be their search steps. In Table 5-5, we summarize the search space for the C-S supply chain. Note that in the process of identifying this search space, we have also located the starting point for our SA algorithm, which is x 1 = [1, 75%, 4, 25, 4 ] The objective value of this initial solution is f[1, 75%, 4, 25, 4 ] = From now on, our job is to find a better solution than [1, 75%, 4, 25, 4 ], with the SA algorithm we described in chapter 4. Table 5-5 Search space of the C-S supply chain Variable Lower Limit Upper Limit Step Size t SL c 65% 85% 2% n Tc K n M Before using SA, we still have two decisions to make: 1) which perturbation algorithm to use and 2) what are the values for the SA parameters. Recall from chapter 4, we have two options for the perturbation and six SA parameters (since we have already decided X 0 = [1, 75%, 4, 25, 4 ], there are five SA parameters left unset). We will try each of the two perturbation algorithms respectively. Then, based on a specific perturbation, we will carry out some factorial design to decide the values for the left SA parameters. After setting the parameters, we can run SA to find the optimum solution for the C-S supply chain. Based on the result of SA, we can then evaluate the performance of the two perturbation options. Now let us review the first perturbation algorithm for the C-S supply chain. 54

68 Step 1: Consider the current solution vector [t, SL c, n Tc, K 1, n M1 ] Step 2: Set a tag for each of the above variables with equal probability Variable t SL c n Tc K 1 n M1 Tag [0, 1) [1, 2) [2, 3) [3, 4) [4, 5) Step 3: Generate a uniform random number r 1 [0,5) Step 4: Based on r 1 select the variable to be changed Step 5: Generate a random number r 2 [0,1) If (r 2 < 0.5) Increment the element subject to its upper limit (see search space) Else Decrement the element subject to its lower limit (see search space) Step 6: Set the new solution vector Based on this perturbation mechanism, we can carry out the factorial designs for the five SA parameters: t i, t f, α, β and π. According to pilot study, we assign two levels for each of the five parameters, as shown in Table 5-6. Table 5-6 Factor experimental levels of SA parameter Factor t i t f α β π Level Level Then, we run the SA procedure (see section ) based on the each combination of the parameter levels. For each combination, we run SA for five times and collect the result in Table 5-7. There are 32 combinations and each combination has five results, therefore it takes our SA to run 160 times to collect all the data in Table

69 Table 5-7 Perturbation one SA parameter experiment on the C-S supply chain t i t f α β π Result 1 Result 2 Result 3 Result 4 Result We use the data in Table 5-7 to run a factorial analysis with Minitab and generate the ANOVA table in Table 5-8. According to the p-values in Table 5-8, there are two parameters which significantly influence the performance of the SA algorithm: t f and β. 56

70 Table 5-8 ANOVA table of perturbation one SA parameter experiment on the C-S supply chain Then we want to decide the levels for t f and β. According to main effect plot in Figure 5-3, SA performances better when t f = 1, and SA performances better when β = 10, respectively. According to interaction plot in Figure 5-4, there s no conflict between t f and β when they are both set to the high performance level. Therefore, we set t f = 1 and β = 10. As for the other three parameters, t i, α and π, we also want to set each of them to a reasonable level instead of setting them randomly. Hence, we need to check their interactions with t f and β respectively to see if there s a non-conflict level choice for them. According to Figure 5-3, when t f = 1 a t i of 1000 contributes better SA performance, when β = 10 a t i of 10 contributes better performance as well. Therefore, there is no conflict. We set t i = According to Figure 5-4, when t f = 1 a α of 0.75 contributes better SA performance, when β = 10 a α of 0.75 contributes better performance as well. No conflict. We set α = According to Figure 5-5, when t f = 1 a π of 10 contributes better SA performance, when β = 10 a π of 10 contributes better performance as well. No conflict neither. We set π = 10. Now that we have decided all the SA parameter values for the perturbation one option, as shown in Table 5-9, we can finally run our SA algorithm to find the optimum solution of the C-S supply chain. We run the SA for ten times and collect all the result in Table As we can see the 9th trial gives us the maximum objective value. And the SA search path for this specific trial is shown in Figure 5-6. As we can see, in order to escape from the local maximum of , several pretty worse solutions are accepted during the first 50 57

71 moves. From then on, our SA search path becomes relatively stable and moves consistently towards a better solution. The final accepted move give as a maximum objective value of The total improvement is = 4.45%. Figure 5-5 Main effects plot (left) and interaction plot of t f and β of perturbation one SA on the C-S supply chain Figure 5-6 Interaction plot of t i, t f, β of perturbation one SA on the C-S supply chain Figure 5-7 Interaction plot of t f, α and β of perturbation one SA on the C-S supply chain 58

72 Figure 5-8 Interaction plot of t f, β and π of perturbation one SA on the C-S supply chain Table 5-9 Parameter of perturbation one SA on the C-S supply chain Factor t i t f α β π Value Table 5 10 Result of perturbation one SA on the C-S supply chain Solution Vector Objective Value 1 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

73 Figure 5-9 Search path of perturbation one SA on the C-S supply chain Next, we want to try the second perturbation algorithm: Step 1: Consider the current solution vector [t, SL c, n Tc, K 1, n M1 ] Step 2: Do the following for each of the variables 2.1 Generate a uniform random number r [0,3) 2.2 Based on r perform any one of the following with equal probability for the variables: Perform Increment Decrement No change Tag [0, 1) [1, 2) [2, 3) 60

74 (a) Increment the variable subject to its upper limit (see search space) (b) Decrement the variable subject to its lower limit (see search space) (c) No change in the variable Step 3: Set the new solution vector Then, we run our SA for another 160 times to collect the similar data as in Table 5-11, except that this time we are using perturbation algorithm two. The results are collected in Table 5-11 Perturbation two SA parameter experiment on the C-S supply chain t i t f α β π Result 1 Result 2 Result 3 Result 4 Result

75 We plug the data in Table 5-11 into Minitab and run an ANOVA analysis. The result is shown in Table According to Table 5-12, t f, β and π are the three significant factors to perturbation two SA performance. Table 5-12 ANOVA table of perturbation two SA parameter experiment on the C-S supply chain According to the main effects plot in Figure 5-10, the insulated high-performance level for t f, β and π are 1, 10, and 10, respectively. According to the interaction plot in the same figure, there is no conflicts between these high-performance levels neither. Therefore, we set t f = 1, β = 10 and π = 10. Figure 5-11 Main effects plot (left) and interaction plot of t f, β and π of perturbation two SA on the C-S supply chain Then, let us decide the levels for the other two non-significant factors t i and α. According to Figure 5-12, when t f = 1 t i s high-performance level is 1000, when β = 62

76 10 t i s high-performance level is either 1000 or 3000, and when π = 10 t i s highperformance level is either 1000 or 3000 as well. Therefore, we set t i = According to Figure 5-9, when t f = 1 α s high-performance level is 0.85, when β = 10 α s high-performance level is 0.75, and when π = 10 α s high-performance level is There s a conflict here. t f and π want α to be 0.85, and β wants α to be Well, 2 vs. 1. We choose to set α = Figure 5-13 Interaction plot of t i, t f, β and π of perturbation two SA on the C-S supply chain Figure 5-14 Interaction plot of t f, α, β and π of perturbation two SA on the C-S supply chain 63

77 We summarize the parameter values in Table 5-13 as decided. And based on these parameters, we run our SA for another 10 times and obtain the result as shown in Table As we can see, both the 2 nd and the 5 th trial give the maximum objective values. The only difference between them is the customer service level, which in the 2 nd trial is 77% and 83% in the 5 th trial. We would like to choose a higher service level so that we choose the 5 th to check its SA search path, as shown in Figure Table 5-13 Parameter of perturbation two SA on the C-S supply chain Table 5-14 Result of perturbation two SA on the C-S supply chain Figure 5-16 Search path of perturbation two SA on the C-S supply chain 64

78 In order to escape from the local maximum of , several pretty worse solutions are accepted during the first 120 moves. After that, our SA search path becomes relatively stable and moves consistently towards a better solution. The final accepted move give as a maximum objective value of The total improvement is = 4.68%. We merge the most right columns, from Table 5-10 and Table 5-14, into Table 5-15 so that we can compare the performance of perturbation one and two side by side. Are their performance statistically different? We take the data from Table 5-15 into Minitab and run a t-test, whose result is shown in Table According to Table 5-16, we can tell that the performance of these two perturbation algorithms on this C-S supply chain is statistically the same. In other words, for a C-S supply chain, we can choose either perturbation one or perturbation two to run the SA, and they will provide the same quality performance in the long run. Table 5-15 Compare the SA results using perturbation one and perturbation two on the C-S supply chain Table 5-16 T-test between the SA results of perturbation one and perturbation two on the C-S supply chain 65

79 Finally, we summarize the optimum allocation of this C-S Kanban supply chain in Table Based on the given parameters, we recommend that the supply chain runs its periodical operation every 0.5 hour, where the service level at the customer shop should be 83%, 3 trucks should be assigned to travel between the customer shop and the supplier, 13 production kanbans should be circulating between the supplier s output buffer and its plant, and 3 machines should be set up in the supplier s plant. Then, based on our estimation, the entire supply chain will be able to generate $ of profit every 0.5 hour. Table 5-17 Optimum allocation of the C-S supply chain C-S-S Supply Chain Now let us optimize the C-S-S supply chain (Figure 5-17). First of all, we assign some numerical values to its parameters, as shown in Table Based on these parameters, we are able to map a specific solution vector to its objective value. 66

80 Customer Shop The 1 st Supplier PK PK PK PK PK PK PK PK PK PK PK PK PK The 2 nd Supplier PK PK PK PK PK PK PK PK PK PK PK PK PK Natural Resource Figure 5-18 One-customer-shop & two-supplier (C-S-S) supply chain 67

81 Table 5-18 Parameters of the C-S-S supply chain According to our model, there are nine decision variables for a C-S-S supply chain, and they are the length of one single operation period, the service level at the customer shop, the number of trucks travelling between the customer shop and the 1st supplier, the service level at the 1st supplier s raw material warehouse, the number of trucks travelling between the two suppliers, the number of machines in the 1st suppliers plant, the number of machines in the 2nd suppliers plant, the number of production Kanbans in 1st supplier s plant, and the number of production Kanbans in 2nd supplier s plant. Therefore, the solution vector of a C-S-S supply chain can be expressed as x 2 = [t, SL c, n Tc, SL 1, n T1, n M1, n M2, K 1, K 2 ] And our goal is find the optimum x 2 and its corresponding maximum f(x 2 ). Next, we need to decide the search space, the search range for each of the nine variables in x 2. K 1 is the first variable we want to focus on. In order to discover the relationship between K 1 and f( K 1 ), the other four variables need to stay constant while K 1 varies. We set t = 1 for the simplicity of calculation, SL c = 80% because of the 68

82 80/20 principle, n Tc = 4 because the single truck capacity is 3 so that 4 trucks should cover the average demand rate of 10, SL 1 = 95% because there s no communication barrier in the same supplier facility so that the service level is expected to be high, n T1 = n M1 = n M2 = 4 for the same reason as n Tc, and K 2 = 30 because 30 is the upper limit for the number of production Kanbans of the C-S supply chain so that it is safe to use 30 to avoid any demand lost. According to Table 5-19, f( K 1 ) begins to increase dramatically when K 1 = 15 and decrease when K 1 = 35. Therefore, we set [15, 35] to be the search range for K 1, and 2 to be its search step. Table 5-19 K 1 experiment on the C-S-S supply chain Now we move to variable K 2. In Table 5-19, f( K 1 ) reach its maximum value when K 1 = 25. Therefore, we set K 1 = 25 and relax K 2. According to Table 5-20, f( K 2 ) begins to increase significantly when K 2 = 15 and decrease when K 2 = 35. Hence, we set [15,35] to be the search range for K 2, and 2% as its search step. Table 5-20 K 2 experiment on the C-S-S supply chain 69

83 Next, variable SL c. In Table 5-20, f( K 2 ) reach its maximum value when K 2 = 25. Therefore, we set K 2 = 25 and relax SL c. According to Table 5-21, f(sl c ) remains high value between SL c = 75% and SL c = 95%. Hence, we set [75%, 95%] to be the search range for SL c, and 2% as its search step. Table 5-21 SL c experiment on the C-S-S supply chain As for SL 1, in Table 5-21, f(sl c ) is at its peak when SL c = 80%. Therefore, we set SL c = 80% and relax SL 1. According to Table 5-22, f(sl 1 ) remains high value between SL 1 = 90% and SL 1 = 99%. And we set [90%, 99%] to be the search range for SL 1, and 1% as its search step. Table 5-22 SL 1 experiment on the C-S-S supply chain Table 5-22 shows that f(sl 1 ) is at a local maximum when SL 1 = 95%. Therefore, we set SL 1 = 95% and relax our next variable t. According to Table 5-23, f(t) begins to increase when t = 0.5 and decrease when t = 1.5. Therefore, we set [0.5, 1.5] to be the search range of t, and 0.1 as its search step. Also, f(t) reaches its local maximum when t = 1. As for variables n Tc, n T1, n M1 and n M2, pilot study shows a safe margin of +1 should guarantee their high performance search range. Therefore, we set [3, 5] to be the search range for all of them and 1 to be their search steps. 70

84 Table 5-23 t experiment on the C-S-S supply chain Table 5-24 Search space of the C-S-S supply chain In Table 5-24, we summarize the search space for the C-S-S supply chain. Note that in the process of identifying this search space, we have also located the starting point for our SA algorithm, which is x 2 = [1, 80%, 4, 95%, 4, 4, 4, 25, 25 ] And the objective value of this initial solution is f[1, 80%, 4, 95%, 4, 4, 4, 25, 25 ] = Our job is to find a better solution. Similar to solving the C-S problem, now let us review the first perturbation algorithm for the C-S-S supply chain: Step 1: Consider the current solution vector [t, SL c, n Tc, SL 1, n T1, n M1, n M2, K 1, K 2 ] Step 2: Set a tag for each of the above variables with equal probability 71

85 Step 3: Generate a uniform random number r 1 [0,9) Step 4: Based on r 1 select the variable to be changed Step 5: Generate a random number r 2 [0,1) If (r 2 < 0.5) Increment the element subject to its upper limit (see search space) Else Decrement the element subject to its lower limit (see search space) Step 6: Set the new solution vector Based on this perturbation mechanism, we can carry out a factorial design for the five SA parameters: t i, t f, α, β and π. The levels of each SA parameter is the same as the ones for the C-S supply Chain. Also, we run the SA procedure (see section ) based on the each combination of the parameter levels. For each combination, we run SA for five times and collect the results in Table Table 5-25 Perturbation one SA parameter experiment on the C-S-S supply chain t i t f α β π Result 1 Result 2 Result 3 Result 4 Result

86 We take the data from Table 5-25 into Minitab and run an ANOVA analysis. The result is shown in Table According to Table 5-26, t f, β and π are the three significant factors to perturbation one SA performance. Table 5-26 ANOVA table of perturbation one SA parameter experiment on the C-S-S supply chain According to the main effects plot in Figure 5-19, the insulated high-performance level for t f, β and π are 1, 10, and 10, respectively. According to the interaction plot in the same figure, there is no conflicts between these high-performance levels neither. Therefore, we set t f = 1, β = 10 and π =

87 Figure 5-20 Main effects plot (left) and interaction plot of tf, β and π of perturbation one SA on the C-S-S supply chain Figure 5-21 Interaction plot of t i, t f, β and π of perturbation one SA on the C-S-S supply chain Figure 5-22 Interaction plot of t f, α, β, π of perturbation one SA on the C-S-S supply chain 74

88 Then, let us decide the levels for the other two non-significant factors t i and α. According to Figure 5-23, when t f = 1 t i s high-performance level is 3000, when β = 10 t i s high-performance level is, and when π = 10 t i s high-performance level is Therefore, we set t i = According to Figure 5-24, when t f = 1 α s highperformance level is 0.85, when β = 10 α s high-performance level is either 0.75 or 0.85, and when π = 10 α s high-performance level is We choose to set α = 0.85 arbitrarily. We summarize the parameter values in Table 5-27 as decided. And based on these parameters, we run our SA for another 10 times and obtain the result as shown in Table As we can see, the 4 th trial give the maximum objective value. Therefore, we choose the 4 th to check its SA search path, as shown in Figure Table 5-27 Parameter of perturbation one SA on the C-S-S supply chain Table 5-28 Result of perturbation one SA on the C-S-S supply chain In order to escape from the local maximum of , several pretty worse solutions are accepted during the first 200 moves. After that, our SA search path becomes relatively stable and moves consistently towards a better solution. The final accepted 75

89 move give as a maximum objective value of The total improvement is = 12.64%. Figure 5-26 Searching path of perturbation one SA on the C-S-S supply chain Next, we want to try the second perturbation algorithm to optimize the C-S-S supply chain 76

90 Then, we run our SA for another 160 times to collect the similar data as in Table 5-25, except that this time we are using perturbation algorithm two. The results are collected in Table Table 5-29 Perturbation two SA parameter experiment on the C-S-S supply chain t i t f α β π Result 1 Result 2 Result 3 Result 4 Result

91 We take the data in Table 5-29 into Minitab and run an ANOVA analysis. The result is shown in Table According to Table 5-30, β and π are the two significant factors to perturbation two SA performance. Table 5-30 ANOVA table of perturbation two SA parameter experiment on the C-S-S supply chain According to the main effects plot in Figure 5-28, the insulated high-performance level for β and π are both. According to the interaction plot in the same figure, there is no conflicts between these high-performance levels neither. Therefore, we set β = 10 and π = 10. Figure 5-27 Main effects plot (left) and interaction plot of β and π of perturbation two SA on the C-S-S supply chain Then, let us decide the levels for the other three non-significant factors t i, t f and α. According to Figure 5-17, when = 10 t i s high-performance level is 1000, and when π = 10 t i s high-performance level is Therefore, we set t i = 3000 arbitrarly. According to Figure 5-18, when = 10 t f s high-performance is 1, and when π = 10 t f s high-performance level is 1. Therefore, we set t f = 1. According 78

92 to Figure 5-19, when β = 10 α s high-performance level is 0.85, and when π = 10 t i s high-performance level is 0.85 as well. Therefore, we set α = Figure 5-29 Interaction plot of t i, β and π of perturbation two SA on the C-S-S supply chain Figure 5-30 Interaction plot of t f, β and π of perturbation two SA on the C-S-S supply chain Figure 5-31 Interaction plot of α, β and π of perturbation two SA on the C-S-S supply chain 79